Suppose that the ring homomorphism $R\rightarrow S$ is faithfully flat ($R$ and $S$ are Noetherian commutative rings). Let $A$ be an Artinian $R$-module. Do we have $0:_S(A\bigotimes_R S)=(0:_RA)S$.
I know this the case when $A$ is Noehterian. Also by purity I know that $\big(0:_S(A\bigotimes_R S)\big)\bigcap R=0:_RA$ for an arbitrary $R$-module A.